Chevalley bases for elliptic extended affine Lie algebras of type $A_1$

TL;DR

This paper constructs Chevalley bases for elliptic extended affine Lie algebras of type A_1, using specific Jordan algebra structures, advancing understanding of integral forms in rank one extended affine Lie algebras.

Contribution

It introduces explicit Chevalley bases for elliptic extended affine Lie algebras of type A_1, a case previously less understood due to rank one complexities.

Findings

Established Chevalley bases for elliptic extended affine Lie algebras of type A_1.

Connected the bases to Jordan algebra structures used in TKK construction.

Provided a framework for integral structures in rank one extended affine Lie algebras.

Abstract

We investigate Chevalley bases for extended affine Lie algebras of type $A_1$.The concept of integral structures for extended affine Lie algebras of rank greater than one has been successfully explored in recent years. However, for the rank one it has turned out that the situation becomes more delicate. In this work, we consider $A_1$-type extended affine Lie algebras of {nullity} $2$, known as elliptic extended affine Lie algebras. These Lie algebras are build using the Tits-Kantor-Koecher (TKK) construction by applying some specific Jordan algebras: the plus algebra of a quantum torus, the Hermitian Jordan algebra of the ring of Laurent polynomials equipped with an involution, and the Jordan algebra associated with a semilattice. By examining these ingredient we determine appropriate bases for null spaces of the corresponding elliptic extended affine Lie algebra leading to the establishment of Chevalley bases for these Lie algebras.